There is a lot of misinformation on here...
A licence is needed, but it comes under the general licence which doesn't need registration and is there by default, so we all have that licence, it only exists as a legal quirk to allow people to have it removed if found to be doing illegal things.
It is not illegal to shoot a pigeon per se but as others have said, there are only certain types of pigeon (basically feral / wood-pigeon) - so look for one with balaclavas over their heads and they are fair game...
you can also only do so for a limited set of reasons, which is basically around protection of crops - a flock of pigeons will strip a field of peas very rapidly, so a lot of farms will have arrangements with folks to shoot them... protecting your solar panels is probably not a likely reason - but it might be if you can show that every other method has failed, but the onus of proof then rests on you...
shooting a gun on your own property is not illegal, but there are boundaries to how you do so... an air rifle has a low range and can be happily shot in a large garden without issue - however, even the compressed air ones and with telescopic sights do not have the power to take out a pigeon - for that you would traditionally use a shotgun.
A more powerful air rifle requires a firearms licence and comes under the same category as a normal rifle - and is less likely to be suitable for a garden
Shotguns come under a shotgun licence - are suitable for pigeons (though still needs a clean shot) but are unlikely to be suitable for shooting on your own land unless you own a number of acres / a farm / an estate.
calling the police on legal use of an air-rifle would be inappropriate - if some of the suggestions above are used it will end up with a firearms response team and the person calling being held responsible for wasting police time...
so - much of what the OP's husband is talking about is not technically illegal - however it is very likely a complete waste of time - and to make it successful make take it out of the realms of being legal...
the suggestion of putting the gun away and finding a company to put up nets is probably the best route...
having said that - a plump pigeon waddling around is a meal on legs - pigeon does taste very nice indeed...