Can I develop diabetes even if I exercise and eat a healthy diet?

The answer, unfortunately, is yes. Genetics is the largest factor affecting whether you are going to develop diabetes. If you have a very strong family history of diabetes, it may be difficult for you to prevent the development of diabetes altogether. The second major risk factor, of course, is obesity.  Diabetes may also be triggered by taking certain medications, especially anti-inflammatory steroids like prednisone or dexamethasone.  These drugs are used to treat many autoimmune or inflammatory conditions, including many forms of arthritis, asthma and other conditions. 
Maintaining a normal body weight, eating right and exercising regularly will help delay the development of the diabetes, even in someone who has a strong hereditary tendency for the disease. For example, someone who is genetically programmed to develop diabetes at a normal body weight when he is, say, age 65, might develop diabetes at 45 if he is substantially overweight. So exercise and proper diet will help prevent the development of diabetes as early as it otherwise would occur.