A dragon cannot use both multiattack and its breath weapon on the same turn
I've used the statblock from an Adult Black Dragon when answering this questions - as its available actions match your description. Not all dragons do have those exact options.
Like PCs, NPCs are generally limited to perfoming one action per turn (exceptions include a Fighter's Action Surge ability and the actions gained via the spell Haste).
When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action, as described in the Player’s Handbook. MM
So, under any normal conditons dragons are thus limited to one action per turn.
In a dragon's action list Multiattack and Breath weapon are listed as separate options and so are mutually exclusive - they are each one action, on their own - so a DM controlling a dragon must choose to have them use one or the other of these options.