Can a lovebird and a conure breed?

Yes they can, but it is not recommended. There are many color mutations in the different Lovebird species, I can see no reason to crossbreed species, when you do you pollute the blood of the different species. These birds get sold and end up with others who unwittingly breed them, the next thing you know it could get hard to get pure birds of a given species. Please don't be part of this problem stay in the species and breed for different colors.