What happens when you select actions using softmax instead of epsilon greedy in DQN? I understand the two major branches of RL are Q-Learning and Policy Gradient methods.
From my understanding (correct me if I'm wrong), policy gradient methods have an inherent exploration built-in as it selects actions using a probability distribution.
On the other hand, DQN explores using the $\epsilon$-greedy policy. Either selecting the best action or a random action.
What if we use a softmax function to select the next action in DQN? Does that provide better exploration and policy convergence?
DQN on the other hand, explores using epsilon greedy exploration. Either selecting the best action or a random action.

This is a very common choice, because it is simple to implement and quite robust. However, it is not a requirement of DQN. You can use other action choice mechanisms, provided all choices are covered with a non-zero probability of being selected.

What if we use a softmax function to select the next action in DQN? Does that provide better exploration and policy convergence?

It might in some circumstances. A key benefit is that it will tend to focus on action choices that are close to its current best guess at optimal. One problem is that if there is a large enough error in Q value estimates, it can get stuck as the exploration could heavily favour a current best value estimate. For instance, if one estimate is accurate and relatively high, but another estimate is much lower but in reality would be a good action choice, then the softmax probabilities to resample the bad estimate will be very low and it could take a very long time to fix.
A more major problem is that the Q values are not independent logits that define preferences (whilst they would be in a Policy Gradient approach). The Q values have an inherent meaning and scale based on summed rewards. Which means that differences between optimal and non-optimal Q value estimates could be at any scale, maybe just 0.1 difference in value, or maybe 100 or more. This makes plain softmax a poor choice - it might suggest a near random exploration policy in one problem, and a near determinitsic policy in another, irrespective of what exploration might be useful at the current stage of learning.
A fix for this is to use Gibbs/Boltzmann action selection, which modifies softmax by adding a scaling factor - often called temperature and noted as $T$ - to adjust the relative scale between action choices:
$$\pi(a|s) = \frac{e^{q(s,a)/T}}{\sum_{x \in \mathcal{A}} e^{q(s,x)/T}}$$
This can work nicely to focus later exploration towards refining differences between actions that are likely to be good whilst only rarely making obvious mistakes. However it comes at a cost - you have to decide starting $T$, the rate to decay $T$ and an end value of $T$. A rough idea of min/max action value that the agent is likely to estimate can help.