Adjective to describe the quality of liking to teach and/or having a talent for teaching?

A natural or born or natural-born teacher.

natural-born adjective : able to do something well immediately or from the very first time -- http://www.merriam-webster.com/dictionary/natural-born


You might consider, pedagogy-minded

  1. the function or work of a teacher; teaching.

  2. the art or science of teaching; education; instructional methods.

Random-House Kennerman Webstee's Dictionary

In order for pedagogy-minded applied linguists to be able to anticipate cases where learners are likely to misconstrue the meaning of a phrasal expression, they need to put themselves in their learners' shoes.

Bilingual Figurative Language Processing