Nevermind, i'm just confusing myself. Now I understand what I misunderstood. The implied volatility surface of a prices of calls generated by a stochastic volatility model will not be constant since the implied volatility is found using the Black-Scholes model. The Black-Scholes model and a stochastic volatility model of course disagree on prices, and hence fixing prices generated by a stochastic volatility model into the Black-Scholes formula to find implied volatilities gives different volatilities for each call.

The implied volatility surface of a prices of calls generated by a stochastic volatility model will only be constant if I use the model itself to find the implied volatilities, obviously.