What is the role of a quint and rescue engine?

Quint is a combined Pump and ALP. if you google CARP this is the UK version.
rescue engine is a combined H/R and Pump.

The rescue engine is a combination of an engine and heavy rescue. The quint is a combination of an engine and platform ladder truck.

Just another tidbit of information, if a call requires say, 3 engines and a rescue, and you send 2 engines and a rescue engine it would be sufficient to complete the call because the rescue engine counts for a engine plus a rescue.