Unclear, but I would guess not.
There are so few Animagi in canon, or examples of Patronuses changing, that we don’t have an explicit example to call upon. I’m going to speculate based on what we know about Animagi an Patronuses.
Your Animagus form isn’t determined by choice, but instead reflects your personality:
When you turn into an Animagus, can you choose what animal you become? Or does this get "assigned" to you?
No, you can't choose. You become the animal that suits you best. Imagine the humiliation when you finally transform after years of study and find that you most closely resemble a warthog.
— J.K. Rowling’s World Book Day Chat (July 2004)
So your Animagus form is determined by characteristics that are “internal”.
The one example we have of a Patronus changing form is Tonks, which is driven by her love for Lupin, which might be described as an “external” characteristic. I would guess that the happy memory she uses to conjure a Patronus is something to do with Lupin, which would contribute directly to the change.
He was, however, reminded of a question he had with regard to Tonks, and who better to ask than Lupin, the man who knew all about Patronuses?
“Tonks’s Patronus has changed its form,” he told him. “Snape said so, anyway. I didn’t know that could happen. Why would your Patronus change?”
Lupin took his time chewing his turkey and swallowing before saying slowly, ”Sometimes… a great shock… an emotional upheaval…”
— Half-Blood Prince, chapter 16 (A Very Frosty Christmas)
Personality tends not to change, but the happy events and memories we have are changing (and hopefully growing) as our life continues. A happy memory, which might motivate a change in Patronus, probably doesn’t affect your inner personality. Hence, I’d be inclined to think it wouldn’t lead to a change in Animagus form.