No this shloka is not from Shreemad Bhagvat Purana.
This shloka is given in Skanda Purana Vol.18 Book VI , Nagar Kanda , Chapter 239 , ( Efficacy of Adoration , Penance etc.) Verse 31-34
जन्मना जायते शूद्रः संस्कारात् द्विज उच्यते।
तथा क्रोधः प्रसन्नता ।|31-34||
31 : A Man is no better than a sudra at his birth .He is called
Brahmana (Twice Born) due to the consecration. The ability to curse
and to bless, the state of being angry and pleased and the status of
being the foremost in all three worlds occur only in Brahmana
This chapter is about adoration of Lord Vishnu is performed by means of Sixteen Upacharas. (Shodasopachara puja) (षोडशोपचार पूजा). This shloka is used in context of Achamana ,Yajnopavita , Dvija's. But the specific chapter talks about Devotion , Chatrumas puja and Panchyatana puja.