President Tayyip Erdogan interrupted a speech in Ankara to offer condolences for "our martyrs" - a term he regularly uses to describe not only combat deaths but also service personnel killed in the course of their general duties.
Erdogan, his office and the ministry did not say what caused the crash, and they did not provide the number of casualties.
Local media said, without citing numbers, that there were both Turkish and Azeri personnel on board.
After a call with Erdogan, Azerbaijan's President Ilham Aliyev said, according to an official readout, that he had discussed "the tragic news of the loss of servicemen" in the crash, which happened near Georgia's border with Azerbaijan.
Turkey's defence ministry said 20 Turkish personnel were on board, including the flight crew.
Turkish and Georgian authorities said they were working to reach the site.
Erdogan was handed a note by his aides as he was finishing a speech in Ankara.
"God willing, we will overcome this crash with minimum hardships. May God rest the soul of our martyrs, and let us be with them through our prayers," he said.
Georgia's Interpress news agency said the plane had crashed in Sighnaghi, a municipality in the Kakheti region in eastern Georgia, bordering Azerbaijan.
It said the incident was being investigated.
US firm Lockheed Martin, which makes the C-130 Hercules that is widely used by air forces around the world, did not immediately respond to a request for comment.
The C-130 Hercules is a cargo, troop and equipment carrier aircraft.
It is described as a four-engine, turboprop military transport aircraft that can make use of unprepared runways for take-offs and landings.
C-130 military cargo planes are widely used by Turkey's armed forces for transporting personnel and handling logistical operations.
Turkey and Azerbaijan maintain close military co-operation.
Video footage aired on Turkish news outlets appeared to show the aircraft spiralling down and leaving a trail of white smoke.
with AP
