To have an axe to grind means to be in a disagreement with somebody. Literally, an axe cannot be put through a grinder and the idiom implies that one is constantly at loggerheads with somebody.
1. The two brothers never seemed to arrive at an amicable solution to the business problems; it looked like they constantly had an axe to grind.
2. The physical trainer and team coach always had an axe to grind; they never seemed to agree about anything with regard to the players' fitness.
3. The teachers had an axe to grind with the accountant over their constantly delayed salaries.