A Russian defence ministry statement said the Turkish vessel approached to 600m (1,800ft) before turning away in response to Russian small arms fire.
The Turkish military attache in Moscow has been summoned to the foreign ministry over the incident.
Relations remain tense over Turkey's shooting down of a Russian bomber.
The Russian SU-24 bomber was shot down by two Turkish F-16s in the Turkish border-Syria area on 24 November. Turkey maintained the plane crossed into its airspace, although Russia disputes this.
And last week, Turkey complained over what it said was a sailor on a Russian naval ship brandishing a missile launcher as the vessel passed through Istanbul. Russia rejected the criticism saying the crew had a "legal right" to protect the ship.
24 November - A Turkish F-16 fighter jet shoots down a Russian Su-24 attack aircraft, allegedly because it violated Turkish airspace. Russia denies the claim and insists the aircraft had been in Syrian airspace. A Russian pilot is killed parachuting from the jet, and a Russian marine is killed in a subsequent rescue mission.
28 November - With Turkey refusing to apologise over the incident, Russia announces a package of economic sanctions against Turkey, including restrictions on imports and travel
6 December - Turkey condemns Russian "provocation" after Turkish media publish showing a Russian serviceman holding a rocket launcher aboard a warship passing through the Bosphorus.
13 December - Russia fires "warning shots" at a Turkish fishing vessel that it says was approaching its warship in the Aegean Sea.
Presidential war of words
Impact on Russian-Turkish relations
Turkey downs Russian fighter jet - what we know
In the latest incident, Russia's defence ministry said its frigate Smetlivy was anchored just off the Greek island of Lemnos on Sunday morning when it spotted the Turkish fishing boat heading towards it some 1,000m (3,000ft) away.
"Despite numerous attempts by the crew of the Smetlivy, the crew of the Turkish fishing boat did not make radio contact and did not respond to visual signals by semaphore or warning flares," the ministry said in a statement.
To prevent a collision, "small arms were fired in the direction of the Turkish vessel from a range that is not lethal".
The ministry said that "immediately after that the Turkish vessel drastically changed course" and passed within half a kilometre of the Smetlivy, still making no contact with the Russian crew.
Russia's deputy defence minister has summoned a Turkish military attache, the statement said. Turkey has not yet given its response to the incident.
Russian military vessels are involved in Russia's intervention against rebels in Syria.
President Vladimir Putin described as a "stab in the back" Turkey's downing of the Russian bomber and has imposed economic sanctions.
Turkey's Foreign Minister Mevlut Cavusoglu warned on Friday that Ankara's patience with Russia was "not unlimited", and accused Moscow of using "every opportunity" to punish Turkey for the downing of the plane.

Officers say the 18-year-old suffered an adverse reaction and later died, after taking MDMA at The Warehouse in Byker, early on Sunday.
A 19-year-old man was arrested shortly afterwards and another two men aged 19 and 21 were arrested on Monday.
Both 19-year-olds have been released on bail, while the 21-year-old remains in custody.
All three were arrested on suspicion of supplying a controlled drug.
Police have urged anyone who was at the venue on Saturday night and took drugs to see a doctor.
MDMA forms the active ingredient in ecstasy pills.
Russia says one of its warships fired warning shots at a Turkish fishing vessel in the Aegean Sea to avoid a collision.
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