If Azerbaijan is expecting a war imminently, it would make sense to divert some to them. The situation in the caucasus is somewhat tense with 3 conflicts currently present, the Israeli Iran conflict. The US Russia conflict and Azerbaijan's own issue with Armenia.
I would they that situation is a lot more tense and imminent than something between Pakistan and India, so perhaps the Pakistanis were willing to get Azerbaijan up and running with aircraft as soon as possible, the statement even says the jets have been integrated into the Azerbaijani Air force.
Idk how they can have been integrated other than a direct transfer rather than waiting for newly built pieces from scratch, which usually takes longer.