"God" is theos (θεὸς), which means "God," "divine," and "deity."  Obviously, the Greeks used this word generically to refer to all divine beings. So, unlike the English word, it doesn't refer automatically to "God the Father," which is the context of our Christian culture. 

In the Gospels, Jesus usually uses the Greek word with an article, so "the God" to indicate when he means "God the Father." This was necessary in a culture that used the word to refer to all types of gods, large and small. The article indicates the one God as opposed to the multitude of pagan gods.  When using his own words, Jesus always seems to refer to the Jewish/Christian concept of God using the phrase "the God" (ὁ theos). 

The first verse, however,  in which Jesus uses the Greek word, however, is Matthew 4:4. This verse is a quote from the Septuagint, the Jewish version of the OT. This reference does not use the article, so the sense of this verse is "a god" or "a divine." However, since the quote clearly refers to God the Father, Jesus is simply following scripture. The next two verses, also quoting scripture, use the standard "the God."

In  Matthew 4:4 the reference to "God" is the Hebrew, Yehova, the unspoken name of God. This is translated most commonly in the Bible as "LORD" in caps, but occasionally as in the verse (Deu 8:3) as "God." Yehova means literally "the existing one," from the root hayaw, "to be, become, or exist." This is the name that sets God apart from everything else. It originally come from the burning bush as "hayaw hayaw," which we translate as "I am that am," but which also means something like "being to be" expressed originally as a verb. (In his original words about himself, God is a verb, not a noun.) As a noun this phrase might be, "the being (or becoming) of existence" or more simply, Yehova, the existing one.

However, Jesus doesn't only use the Greek theos, to refer to the Christian God. For example, in John 10:35, Jesus uses the plural, "gods" (Θεοί) to refer to ordinary men. Again, he is quoting scripture here (Psalm 82:6).  He doesn't use any articles, "the gods," because he isn't referring to the one God.

We see the dropping of the article again is Mark 12:27 ("He is not the God of the dead"). This time the verse is in his own words, but Jesus is referring to "a god of the dead," a pagan concept. This idea is lost in the English translation, which adds the article, "the," which simply isn't there in the Greek.