Why isn't there A ? " I got involved in trouble."

Yes, in this particular case the sentence is wrong. The correct expression is 'I got into trouble'.
Trouble is a noun, here are several meanings for you:
1. A state of distress, affliction, difficulty, or need: tried to console them in their trouble; got in trouble with the police.
2. A distressing or difficult circumstance or situation: I've had troubles ever since I took this job.
3. A cause or source of distress, disturbance, or difficulty: The new recruits were a trouble to him.
4. Effort, especially when inconvenient or bothersome: went to a lot of trouble to find this book.
5. A condition of pain, disease, or malfunction: heart trouble; car trouble.
6.
a. Public unrest or disorder.
b. An instance of this; a disturbance.