I think that conventionally implicit costs are positive but if for example we consider a negative interest rate as an implicit cost, the implicit cost would be negative. Is this true? Can implicit costs be negative?
We can define a (positive) cost to be simply a negative benefit.
In which case, a (positive) benefit is simply a negative cost.
So yes, in general, costs can be negative.
Note though that in economics, it doesn't usually make sense to speak of costs in isolation or benefits in isolation. Instead we usually look at the net benefits (i.e. all benefits minus all costs) or net costs (all costs minus all benefits).
So to use your interest rate example, usually when one buys say a 10-year bond, the benefit one gets in return is the positive interest rate. However, countries like Germany and Japan have issued bonds with negative interest rates. So, it seems silly that anyone would buy such bonds -- with negative interest rates, one is clearly getting a (negative) benefit and hence incurring a cost. However, this cost must be compared to other costs -- for example that of safely storing some sum of money. When the proper cost-benefit analysis has been made, we will usually find that for the buyers of these German and Japanese bonds, the net benefits are positive (or equivalently, the net costs are negative).