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Should I relocate a pigeon?
Sadly the answer in this case is “no”. One of the things about pigeons that has made them so useful to the human race in time gone by, is that they can find their way home. Pigeon navigation methods are still being studied, but scientists are now working on the theory that they can map their nesting areas with low-frequency sounds. This is what helps them to return to that one spot, even if they are released hundreds of miles away. Yes, pigeons do get lost from time to time, no navigation method is fool proof, but theirs works more often than not.
As in most cases of dealing with problem birds, deterring them is much better than trying to get rid of them once they are happily ensconced in your eaves. Seal off any access to suitable nesting sites. There are many ways to do this. Wire mesh can easily be inserted into or over the problem area, making it impossible for the pigeon to return. Also removing food and water in the area, or making it very difficult to reach, will cause them to simply leave. Doing all three will most certainly cause even the most stubborn pigeon to simply relocate rather than face the hassle of surviving in those circumstances.
If for some reason this is not an option, then there are other ways to get them to leave. The market is full of deterrents including fake falcons and owls that can be set up outside their preferred nesting spaces to make them think that a predator is on their trail. Laying down bird spikes will also help. These spikes are clearly visible to the birds and will stop them from being able to roost in that area. They are also a good people deterrent. With the new research suggesting that pigeons use low-frequencies to navigate, it might be worthwhile to get a low -frequency pest deterrent unit. There are many on the market that claim they repel even the most persistent pigeons.
As a last recourse there is the option of killing them. This is most effective, but is also questionable. Poison should never be used. The risk of contaminating and killing another species is too great, and shooting them is only legal in rural areas, due to safety issues in densely populated areas.