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.
Sample Usage
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.
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