Yes you can brush your teeth after having a deep cleaning of your teeth from the dentist.
To keep your teeth clean and your teeth and gums healthy you need to continue your daily brushing of your teeth as well as flossing of your teeth to keep the teeth clean and healthy.
Your dentist should tell you that you need to brush your teeth twice per day at least day and flossing daily to keep oral bacteria at bay and encourage your gums to continue healing.
If you don't continue brushing your teeth then bacteria can form again which would defeat the purpose of having the deep cleaning.
So if you already are brushing your teeth daily and flossing then you should and can continue doing so after a deep teeth cleaning.