Technically, they could, but that would require a complete abrogation of all norms. We have at most 10 session days remaining. So:

1. Articles of impeachment need to be drafted (probably already long-since done)
2. Are you going to debate in the judiciary committee, our just skip that step? You just passed the house rules, so are you going to vote to change the rules or skip that step too?
3. The President SHOULD have the opportunity to defend himself. Are you going to skip that step too?
4. Assuming YES to skipping all steps, you would have the articles raised on Monday Jan 11 and voted on by Jan 13.
5. You would typically have a committee negotiate the rules for the Senate trial. Typically this is 2-3 day process at least. Would you just throw some rules together and effectively skip this step?
6. You would typically provide the President with time to prepare their case. Maybe we skip this step? Again, you'll need to either violate the rules of the Senate or vote to change them urgently.
7. You would need to swear in all the senator scum
8. You would typically allow for the presentation of the case against the President and allow the President to present his case. would you skip this step?
9. You would then provide time for deliberation and debate. Would you skip this step?

The only way you have a conviction within 10 days is if the lethargic cowards in the Senate have such a fire lit under them, that they are ready to be seen as rushing this through with great urgency. What justification could there be for such urgency? Technically, it is all possible, but I doubt the coup is that well organized and I doubt the conspirators are ready for what would happen back home.