An air ambulance came to rest on the grass at Aberdeen Airport, a Scottish Ambulance Service spokesman said.
A statement on Twitter from the airport said: "Flights suspended for around two hours as a result of an ongoing incident. A further update will follow."
A Police Scotland statement said: "We are aware of the incident but on this occasion there was no requirement for police involvement."
No-one has been injured, officials said.
Flights from Aberdeen are being diverted to Edinburgh and Glasgow.
A spokesman for the ambulance service said: "An air ambulance plane on a routine non-emergency transfer from Shetland appears to have come off the runway at Aberdeen while taxi-ing after landing.
"There were no injuries to anyone on board and the patient has been transferred to hospital."
The incident involved an aircraft operated by Farnborough-based Gama Aviation, it is understood.
Fire crews and paramedics were alerted to the scene but none of those on board the plane were hurt as a result of the incident.
There were four crew members and one patient on board at the time.
The plane was transporting the patient from the Shetland Islands to Aberdeen, with a view to taking the patient to the city's Royal Infirmary.
The incident happened at around 11.50am today, Aberdeen Airport confirmed.
It is not yet known whether the aircraft was damaged as a result of the incident.
The incident was thought to have been stood down within half-an-hour but the operation to move the aircraft from the scene has led to the temporary suspension of flights.
The Scottish Fire and Rescue Service confirmed crews were called to the scene at 11.51am.
One fire engine from Dyce "stood by" when it got to the airport, a spokeswoman said.
Other crews initially called to the scene were turned back on their way to the airport.
All fire crews were stood down at 12.18pm.
Meanwhile, thousands of travellers are enduring flight disruption due to a one-day strike by French air traffic controllers.
Ryanair had to axe more than 70 flights, while easyJet was forced to cancel more than 50 services due to the action which started at 5am UK time and was due to end at 7pm.
Both carriers said they had been asked by aviation authorities in France to reduce their French flights by 30%.
EasyJet said: "We will cancel at least 50 flights to and from Paris Charles de Gaulle, Paris Orly, Basel, Bordeaux, Lyon, Marseille and Toulouse.
"The strike will also impact all flights that over-fly French airspace. More than 60% of our flights operate through French airspace and so there is a risk of delays and late notice cancellations depending on the scale and effects of the strike action.
"Therefore, flights from the UK to destinations such as Spain, Portugal, Italy, Cyprus, Greece and North Africa face the threat of disruption."
A number of the Ryanair cancellations involved services within continental Europe, but the axed flights also included some between UK and Ireland airports and French destinations including Paris, Biarritz, Nimes, Carcassonne, Bezier and Bergerac.
Some Ryanair services to and from Girona in Spain were also cancelled.