This one depends an AWFUL lot on your local building code requirements, and on your yard surface conditions. Some jurisdictions allow flexible plastic tubing runs as shallow as 12 inches, others require as much 48 inches deep. Also, some require flexible plastic gas line be laid in PVC conduit or even steel casing if within a certain distance of the house and where passing under any "hard" surface like sidewalk or driveway. Some allow flexible tubing, some require black iron pipe, etc. The reason for these requirements is partly to provide more protection against puncture from something driving over it, and partly to provide an open passage for any leaking gas so a gas pocket can not form near the house or under an impermeable surface.
There are also code restrictions on how close natural gas lines can come from other utility lines - both horizontally and vertically, so you need to locate other utilities to ensure proper clearance for your code area. Usually not extreme - typically a foot vertically and 1-2 feet horizontally from electric, for instance, which is the most extreme requirement. Can be an issue if you are crossing a buried power feed to the house, or have yard lighting or a pool or such.
Generally, I would say about $150-250 or so for materials depending on how much is through or along the house, and how much under yard, and about $150-300 for plumber labor if you dig the trench where they say to lay it so it meets code. If they trench, depends on your soil conditions - could be additional $100-300 range in soft loamy topsoil without tree roots, to $1000+ range if in solid granite or you have lots of large tree roots to cut or tunnel under. Add from zero to maybe $250 for permit depending on where you are located - some cities have a very high minimum permit fee, especially on gas connections. Note in most areas a specific class of sand or gravel backfill is needed for the bottom 4-8 inches of the trench excapsulating the pipe also - not just any excavated material.
I am assuming your BBQ is already plumbed for natural gas - you cannot run natural gas in a propane grill or BBQ without changing regulator and burners - can be $50-100 range for conversion kit, if available for your grill. Also, bear in mind that propane has about 2-1/2 times the heating capacity as natural gas for the same gas flow rate, so many people are sorely disappointed converting to natural gas because their grill does not heat up as well or as quickly as it used to without an excessively large flame.
Another factor is the odorant added to propane and natural gas can give anything cooked in direct contact with the flame a waxy or chemical taste - from mild to pretty strong, depending on gas source - so depending on your locale and what they use to make the gas smell, or if it has a high natural hydrogen sulfide content like some areas that use locally sourced gas that has not been fully cleaned up to interstate pipeline standards, could destroy your plan. Talk to neighbors with natural gas grills before committing to this.
If cost is getting out of hand, SOME jurisdictions allow use of a flexible hose from a quick connect with shutoff valve on the house wall, run to a BBQ quick connect, and you just string out the hose to the BBQ for use, and when done disconnect and hang up in the garage till needed again. A 1/2 inch (big enough for fairly large grill - probably about 75,000 BTU or so) 25 footer is about $60 with quick disconnect fittings at Amazon. Would still have to pay for grill conversion if needed, and for the piping from gas line to nearest point on outside wall that meets code, and for shutoff valve and valve box and quick connect on the pipe, but would save all the trenching and pipe to the grill so could be comparable in cost or even cheaper in some strict code areas or if your area has hard digging.
If you run the numbers, probably about 90% of people planning this conversion end up not doiong it and stay with propane - you have to be a pretty serious barbecuer, or have other use for the gas line like for outdoor patio radiant heater or pool heater, to justify the installation cost with the maybe 4:1 to 6:1 typical difference in price for the gas.
The contractor category you would want to Search the List for would be Plumber.