The y is for “il y a une bonne bagarre”, the compter is “to count on” with the nuance of anticipation, prevoir in french.

Therefore : You should not count on it (on any good bagarre).

Faut pas trop y compter is a shortcut for

Il ne faut pas trop compter là-dessus.

faut is third person of the defective verb falloir.

This translates:

One shouldn’t count too much on it.

or

One shouldn’t rely too much on it.

So two things in your question:

1. faut is a standard truncation for il faut, a necessity modality marker; in this context => you shouldn’t
2. y compter stands for compter sur ça => count on it

Dropping the impersonal il happens with other verbs:

• faire: fait froid, fait suer, …
• y avoir: y a rien, y a pas photo, …

But not all, with meteo verbs, it’s not common to drop the pronoun:

• il pleut => [iplø] but not [plø]

“Faut pas trop y compter” has a simple correspondence to English:

• “Faut pas” – must not
• “trop” — too much
• “y” — may depend on context but in this case, ‘on it’ or ‘on that’, i.e. “what you just said”
• “compter” — count

So “Must not count on it too much”.

More idiomatically,

• I’d really like a good fight
• “[you or we] Can’t count on that!” or “You’d better not count on that too much”, implying, “That good fight you’re counting on might not happen”