A 'chit' is a small piece of paper, often used in basic accounting, to say that someone owes another person a sum of money.
In this case, it means 'calling in a favour'. If you call in a favour, you are asking someone to repay a good turn which you did for them on a previous occasion. In other words, the character is saying 'Are you asking me to do this because I'm your friend, or because I owe you a favour?'.
And just a bonus tip: Your question should have said, for example 'I came across this phrase in the serial Suits'. The phrasal verb 'go into' can't be used with this meaning.
I hope that helps.