Based on the level of respect or formality, you can say "Do you mind if I speak to you in Korean?" in Korean in 3 ways:

• Informal: nae-ga hanguk-eo-ro mal-haedo gwaechan-a? (내가 한국어로 말해도 괜챦아?), nae-ga hanguk-eo-ro mal-haedo gwaechan-ni? (내가 한국어로 말해도 괜챦니?)
• Formal: jae-ga hanguk-eo-ro mal-haedo gwaechan-ayo? (제가 한국어로 말해도 괜챦아요?), jae-ga hanguk-eo-ro mal-haedo gwaechan-seupnigga? (제가 한국어로 말해도 괜챦습니까?)
• Very Formal: jae-ga hanguk-eo-ro malsseum-deuryeodo gwaechan-getseupnigga? (제가 한국어로 말씀드려도 괜챦겠습니까?)

- you (subjective): neo (너), neo-neun (너는), dangsin-eun (당신은), dansin-ggeseo-neun (당신께서는)
- to mind: gwaechanta (괜챦다)
- to speak: mal-hada (말하다)
- Korean: hanguk-eo (한국어)
- language: eo (어), eoneo (언어)

Please refer to How to pronounce Korean Alphabet for the English pronunciation in this answer.