If the sentence is "Summer's just begun", then that makes sense. Summer's = summer is / summer has. You know it's "Summer has.." (past tense) because of the following word "begun".

It also might be a throw-back to Olde English. German still has the thing where some of their verbs use as an auxiliary the equivalent of "have" (haben) and some use the equivalent of "be" or "is" (sein or ist). It used to be this way in Olde English also and is still apparent in the works of Shakespeare. I think probably an Englishman or Englishwoman of today who is very educated in literature might speak this way once at times when they are feeling their literary heritage. I'm sure it was more common in the past.

In the movie，the turn of seasons need fairies’ help. The fairies use magic to change the seasons. The above comments represent only individual opinions. : )

"Summer's just begun" was the way a lot of English poems started .It was also a song from the movie High school musical "A real summer has just begun! Let's rock and roll and just let go, feel the rhythm of the drums We're gonna have fun in the sun Now that all the hard work, work is done! Everybody, one for all and all for one! All for one, one!"

I think you meant Summer has just begun.