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
the function or work of a teacher; teaching.
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