I have a debt for $2600 that I owe to collection agency global acceptance credit. If I pay them monthly payments less than what they want can they still take me to court?

Yes, they can and proably will because if they get a judgment they can attach your wages and freeze your bank accounts If they agree to a less amount GET IT IN WRITING! How far are you behind in payments?

A judgement gives them leverage. Have you started making payments? What are the full circumstances?

Yes, if you are late or in default of the loan, they can actually even take you to court of you pay them what they ask for ( should not, normally don't but could since you are in default)

But if you send less than they ask for, they will most likely go to court unless this is a short term thing and they know about it and have agreed to it.