Is the termination codon included as part of the coding or non-coding region of a gene?

It is non-coding, but this does require some explanation.

The three stop codons (UAA, UAG and UGA) are transcribed into complementary mRNA codons during transcription.

However, during translation, none of these codons are translated into an amino acid. They merely signal the end of translation. (Instead of the relevant tRNA, a protein called "release factor" binds to the stop codon, and causes the protein to be released.)

Hope that helps!