Wow. Where to begin...
First, you're creating a new Timer on every cell click. THAT'S REALLY BAD! Timers are a finite resource in Windows and you're not destroying the ones you create, so you're eventually going to have dozens of Timers running, all doing the same thing!
Using a Timer is still possible, IF AND ONLY IF the user doesn't get the chance to do any editing in the grid you're trying to refresh. Refreshing it while editing will cancel the users editing, right in them middle of them doing it.
But, using a Timer for refreshing something from the database has a downside, scalability. The more clients you have refreshing on a Timer the more queries you're going to have hitting the database. It'll get to the point where the clients are just flooding the database with requests for the same data.
Also, if you're retrieving a large number of records to shove in the DataGridView, you're putting a larger load on the database and on your clients. A data bind is not a lightweight, quick operation.
To solve that problem, you'd have to use paging or virtual mode in the grid. That complicates things.
Second, look at the signature of the timer_Tick event handler you're trying to wire up:
private void timer_Tick(object sender, DataGridViewCellEventArgs e)
The event arguments are for a DataGridView event, not for the Timer Tick event. That's why you're getting the exception you are. The Timer Tick event just uses the generic EventArgs argument:
private void timer_Tick(object sender, EventArgs e)
To use a Timer, you'd put the timer on the form and double click it to get the event handler started for you. Do your database data retrieval and grid refresh from in there. Start the Timer when your form is shown and you're done. There is no need to create a new timer on every cell click.
There are other options to using a Timer but the code becomes more and more complicated, such as using SQL Server Notifications (old), server-side components, caching and subscriptions, services, blah, blah, blah, ... nothing something for the "new to this" crowd.