This will be a short post, but wanted to be sure this was addressed.
During a Q&A on Twitter, J.K. Rowling was asked what Hagrid’s patronus would be. Her response wasn’t too surprising.
Hagrid couldn’t produce a Patronus. It’s a very difficult spell. https://t.co/TRDVNvSEdR
— J.K. Rowling (@jk_rowling) August 21, 2015
Honestly, it really isn’t that surprising. It has never been officially stated, exactly how old Hagrid was when he was expelled for being falsely accused of opening the Chamber of Secrets. As part of being expelled, his wand was snapped, but thanks to Dumbledore was able to keep it concealed in a pink umbrella.
Regardless of not knowing Hagrid’s age at expulsion, we don’t know when the Patronus charm is actually taught. Harry was taught this at the age of 13 due to the extreme effect the dementors had on him. He then taught the members of DA at the age of 15, quite successfully, but it’s never stated when, or if, this is normally taught at Hogwarts. According to Remis Lupin, even many qualified wizards have difficulty conjuring it.
Taking all this into consideration it’s really not surprising that Hagrid, who never completed his studies at Hogwarts, would not know how to conjure this incredibly difficult spell.