To speak fast. To speak slow
I found these examples in a text-book today. I am amazed at the fact that instead of adverb after “speak” adjective is used.
And one more question: can I use quick(ly) instead of fast?
‘To speak quickly’ and ‘to speak slowly’ are the standard, correct terms which could have been used.
Although you will come across ‘speak fast’ in semi formal settings, ‘speak slow’ is the most colloquial and ‘incorrect’ of the options.
Thank you dear Beeesneees 
You are so attentive