I think the simplest answer is that the Bellman eq. relates the value of the current state with its neighbors. Q learning is a method for learning the best actions in each state (a policy) using the Bellman eq.

The bellman equation is the general mathematical tool for MDPs via dynamic programming and to find the optimal policy. It says that the value of a state is equal to the optimal action (=the action with the maximal reward) plus discount factor time the value of the succesive state, the action takes us to. To get the value of the next state the equation is applied again. This is done recursively until the final state is reached. For stochastic processes the probability to get with a specific action to a certain state has to be considered.

In Q-learning the bellmann equation is used with the modification that instead of evaluating the value of state, we ask for the value of a state-action pair (=Q-value). The goal of RL is to determine the Q-values and the derieve the optimal policy.

In another way to think about this, the Bellman equation requires knowledge of the environment's transition matrix/function (i.e. "a model of the world"). Q-learning does not require any knowledge of the transition matrix. Hence, Q learning is often known as "model-free".