The system is designed for every-player on the same side. The moment you move to PVP, you’re outside the design space for this flavor of 2d20.
There’s no answer in current rules, because the action taken is outside the scope of the design.
JCoM uses separate pools by player; in that version, PVP is much easier to adjudicate, because it doesn’t presuppose teamwork the way STA does.
I guess I wasn’t clear. Think team A being enemy Klingons (GM run) and team B the players.
Official word from Modiphius is that you cannot do a swift task with momentum gained from an opposed task in HtH combat. So, your assumption was correct, mattcapiche.
But there would be no issue with using it for, say, adding to the resulting damage!
Yes, I agree. I think anything that is immediate or repeatable is allowed. However, Modiphius only commented regarding Swift Task.
I would like to ask extra question on PvP and momentum. I know Star Trek is mostly about player cooperation, but there may be a scene where PC have conflicting agendas, for example one PC wants to break Prime Directive and another wants to protect it.
It does not really matter if they try to argue and dispute with each another, or they just punching their way out of this situation. Still there is a question:
Would you allow a use of group Momentum in pvp test? Does this kind of test generate extra momentum if one PC wins?
Generally, I think this falls partly under the question of ‘Do you want to allow your players to actively fight each other whenever they disagree, and potentially have one player purposely try to kill another player’s character?’
If you don’t really care one way or another, then allowing PCs to fight one another should follow the rules for every other form of conflict. If one character generates more successes than the other, then the winner generates momentum.
If, however, you want to discourage players from fighting each other (either non-lethally or lethally) then do not allow momentum to be generated when fighting other player characters.
Honestly, I am less concerned with players fighting each other, since there are Red Alert pvp rules, which may be adapted to this kind of situation.
But consider Starfleet away team landing on one of Maquis’ worlds with orders to find and arrest one of famous terrorist. PC conduct investigation and find where is hiding place for Maquis group and now there are planning the ambush.
During this process one PC realizes that she has a family members involved in terrorist group. She secretly contacts them and information she gets change somehow view on whole situation. She considers that Maquis are not really bad and their ‘terrorist’ activity may by easily explained as a self-defense against some aggresive moves from Cardassians.
Now we have two agendas in our team: part of them want to follow Starfleet orders and really do not care about nuances of political situation and another part of team want to suspend the operation and investigate more, maybe even try to change orders.
Having this kind of conflict may be settled over a table between players and of course there should be a lot of debate and discussions, it can be ruled as a roleplay, where we all hear arguments from both sides and decide what to do next.
But from my experience I should be prepared that players want to settle this mechanically - similarly to fight, as an opposed check. Some of PC have diplomatic or persuasive talents (using momentum or threat), they may want to use determination even. Of course result of this task will be followed by roleplay, but I would like to know if there is mechanical support for this kind of challenges.
If you allow the use of Group Momentum, the first player could theoretically use up all Momentum for their roll to buy dice and create various Advantages. This means the second player is starved of Momentum. Also, do you really find it a good idea for two players building up Threat just to win a PvP?
Generally, there is nothing wrong with PvP, but I would let these play out separately from the Group Momentum and Threat pools. I may allow adding Threat for Talents that require it. If players generate Momentum, I would save it in a temporary personal pool that mirrors the group pool. Additionally, if the situation warrants it, I will add Threat if the PvP goes on for too long.
Personally, when it comes to characters disagreeing with each other, I wouldn’t allow a binding roll for someone to persuade the other, as it doesn’t encourage players to play their characters. Generally I would have the characters make rolls to see how persuasive they argue their point, and then have the players take that into account when they make the decision on what a character would do.
In terms spends, you could consider allowing a threat spend in place of momentum (still letting it trigger the talents). That threat could be seen as simulating the increased danger caused by the discord amongst the crew
I like idea that conflicting sides roll for persuasion and then all players and GM decide how to apply this result in roleplay, since conflict is between two PC, not two players.
Definetly. It discourages PvP mechanics-wise. I will definetly allow my players to engage in PvP. After all, Roleplaying is about freedom of choice. But I will pile up Threat on the way.
In Star Trek, cooperation wins. As long as I GM, this will hold true to my table.
Good point. Then the standard Threat mechanic should work well.
Actually, I thought about having them pay with Threat to use even existing Momentum for PvP. In my opinion, this is more fair than having the player that is already going second also being the only one paying Threat for their Momentum.
I like the idea of using Threat: it appears to be within the spirit of the game as intended by the writers.
If you think about it, disunity amongst the crew is a threat to the success of the mission, and that’s the kind of thing the Threat mechanic was designed to reflect.
Maybe allow Momentum if all the players agree?
Yes. And, also, add Threat as GM if it fits the story (so the group can counter this with Momentum out-of-character).
For a good example, how to handle PvP and Momentum/Threat spends, cf. Shield of Tomorrow, the episode where T’Lan rage-attacks Sage.
do you remember which episode Shield of Tomorrow?
PVP
What thee are about to see comes down from the time of the beginning, without change. This is the Star Trek heart. This is the Star Trek soul.
Unfortunately, no. But it was one of the last episodes. There was fanart on Twitter, dated 7th June 2018, that dealt with the fight. So it was probably 3.22 (Insights). I recall that the rage hat something to do with an orb-induced vision, this would fit within the story of SoT at those episodes. But I cannot nail it down, further, sorry.
But, the whole show is very enriching to watch and Eric Campbell is one hell of a GM even experienced GMs can learn a lot from. So I recommend to just watch the show from the first episode; the fight will come eventually. 
The Conan game includes a little section that covers PvP interactions. There it says that in PvP, players cannot use Momentum from the group pool nor can they add Momentum to it.
To decide which player begins, each bid Threat (secretly). The player with the highest amount wins and that amount of Threat is added to the Threat pool. Ties are resolved by rolling a d20.