I feel that it sounds weird although it should be ok.
Is "I put it inside quotes" better? Why?
In this case, in the dialects I'm familiar with, in is more natural and idiomatic than inside.
Now, the why is a gut feeling rather than anything I know about from theory. There's a natural tendency for any expression that is used frequently to be shorter (that is confirmed in theory). People prefer things to be shorter if possible. Both in and inside make sense here, but in is shorter, but to write and to say, and therefore preferable.
Inside is more likely to be used in cases where there's an emphasis, or a contrast with outside.
This one is inside the quotes, while that one is outside.