I don't think that "zero complaints per year" is necessarily the best goal to set, especially if you really haven't defined what a complaint is, how many complaints there are now, or what the severity of the complaints may be. It becomes too easy to play with the data to make things look the way we want them to look. I do, however, agree that complaints should be tracked and analyzed to help the organization to improve.
You should probably start with a definition of what a complaint is. An example is the medical device definition of customer complaint:
"written, electronic or oral communication that alleges deficiencies related to the identity, quality, durability, reliability, safety or performance of a medical device that has been placed on the market."
You could probably adapt it to work for your organization. Something like "written, electronic or oral communication that alleges deficiencies related to the quality, durability, reliability, useability, safety or performance of an installed system or to the timeliness of installation or effectiveness of training provided."
Having the term "complaint" clearly defined up front will help to avoid the question of whether an exchange with the customer was really a complaint or just a normal part of doing business. The complaints should realistically only include items showing negative perception on the part of the customer.
Examples of customer complaints:
- I thought this was supposed to do XX, but it doesn't.
- You promised XX% uptime, but we've only been getting X%.
- We needed this installed last month, but we're still waiting on XX.
- The system you designed/installed is too hard to use.
- Your system sucks (unless of course it's a vacuum system).

- As soon as your trainer leaves, we're lost as to how to run this XX.
Examples of non-complaints:
- I have a question about...
- How much would it cost to add XX to the system?
- Can you help us to solve a problem with XX (something you didn't design/install)?
- Can you give us a quote for...
The complaints should be analyzed to determine what caused the actual or perceived problem. Even if the problem wasn't your company's fault, some analysis should be done to see why the customer thought it was. This could lead to more clearly defined statements of work etc. to avoid the customer thinking that your company didn't do what was promised.