I don't think so. As I understand it, gravity and acceleration are equivalent, if you aren't experiencing any acceleration then time dilation won't occur. (Assuming we are not traveling near the speed of light)
Think about it: when you are in free fall you feel 0 acceleration. You appear to be accelerating relative to the ground-- but you're actually motionless in an "inertial reference frame". (Similar to how the astronauts on the ISS don't "feel" acceleration despite accelerating rapidly relative to the earth.)
The "force" of gravity is often modeled as "gravity pulling you down" and the ground "pushing you back up". This works mathematically, but isn't quite logically consistent.
In reality, on the ground you're in a region of warped spacetime, so you feel constant upward acceleration despite not actually accelerating. (Thinking of this another way, standing on earth feels identical to being in a far away spaceship accelerating at 9.8 m/s².)
This is also why time "speeds up" near more massive objects. (Separate from "acceleration".)
We're so used to gravity this it doesn't seem weird. But when you consider the fact free-fall is when you're not accelerating... well pondering that from many angles is what ultimately led Einstein to his model of relativity.
(This is me trying to condense what could be a 10 minute explanation into a few sentences, so apologies if it's not particularly clear.)