Technically, it would not be Q-learning if you used Monte Carlo returns, because Q-learning is specifically a temporal difference (TD) method.
However, your intuition about the estimates being interchangeable is correct. There is nothing preventing you in principle from swapping out the TD return estimate for a Monte Carlo sample to generate data for training a neural network that is approximating $Q(s, a)$. In some ways it is a better estimate, as it will be unbiased. If you stop using bootstrapping, then the learning process is theoretically more stable and less likely to diverge.
There are some caveats:
Experience replay becomes less practical. You could store the final return from each $(s,a)$ pair, but it will quickly become off-policy to a degree that it could not be used. Some extensions to DQN do use a small n with n-step returns and no correction. However, you have to be careful with this, and full trajectories off-policy would need better correction - many would be useless, especially at the start of learning.
Without experience replay, you will need more samples of the current policy output before training. Training on a batch generated from just a single trajectory could easily result in poor learning performance, because records would be closely correlated with each other. So you would need to borrow ideas from CEM or A2C where you generate many trajectories for each training batch.
These issues could make a Monte-Carlo based deep RL slower to train than DQN. However, perhaps the extra stability of the approach may appeal for a specific problem.