The borehole, near Belcoo, will be to collect rock samples and will not involve fracking.
Tamboran Resources said drilling will last about 30 days.
If the samples show commercially viable levels of gas, Tamboran will begin a process that could lead to fracking.
The company said the borehole will be about 15cm across and 750m deep.
Residents from the Belcoo area are planning a protest on Monday evening after they said drilling equipment has been installed at a local quarry.
Donal O'Cofaigh, a resident and one of the organisers of the protest, said:  "There has been absolutely no consultation whatsoever with us despite major concern by local residents and farmers at the potential risk associated with drilling and fracking.
"The first we knew that drilling equipment was being installed at the Acheson & Glover quarry was at 5am this morning, when local farmers witnessed the cavalcade of security escorting the trucks."
Tamboran are intending to drill using "permitted development rights", which means the firm does not need to apply for planning permission.
However, the Environment Minister Mark H Durkan has warned those rights may not apply if the work is likely to have "a significant environmental impact".
He said he has instructed officials to "consider carefully" whether or not this is the case.
Mr Durkan added that he will make a statement about the Environment Department's decision "in due course".
In fracking, a mixture of water, sand and some chemicals are pumped into a well under high pressure to force the gas from the rock.
In 2011, Tamboran was granted a petroleum licence for an initial five-year period from the Department of Enterprise to explore for natural gas in Fermanagh.
The collection of rock samples is a requirement of the work programme set out by government within the licence.
Tony Bazley, director of Tamboran Resources, said this work amounts to "fact-finding, not fracking".
"Tamboran's intention at this stage is only to verify that the elements necessary for natural gas and its recovery are contained within the shale in County Fermanagh."
He said that the test drilling should quickly show if there is enough gas to make extraction commercially viable.
"If we find a significant gas resource, it is likely we will continue to the end of the licence term in 2016.
"Before the end of the licence term, a single site would be tested for natural gas if given planning permission."
He added that this is "still a long way off and ultimately any decision to use hydraulic fracturing will be made by the government of Northern Ireland".
The prospect of fracking has provoked controversy in Fermanagh, with some local people concerned that it could cause environmental damage.
The former environment minister, Alex Attwood, has said Northern Ireland should adopt an "enhanced precautionary approach" to fracking.
He said that was because it had not yet been proven to be a safe and responsible source of energy.
The company that plans to use the controversial fracking technique to extract natural gas in County Fermanagh intends to drill an exploratory borehole in August.