There is no simple way and you have to make correlation assumptions.
For instance say you have a volatility surface for $\text{EURUSD}$ and another volatility surface for $\text{USDJPY}$ and you want to build a volatility surface for $\text{EURJPY}$.
You start from the observation that a call with maturity $T$ and strike $K$ on $\text{EURJPY}$ with payoff in JPY is equivalent to an option to exchange a quantity of $\text{EURUSD}_T $ USD for a quantity of $K \text{JPYUSD}_T$ USD so that
$$
D_{\text{JPY}}(T)E^{Q_{\text{JPY}}}[(\text{EURJPY}_T - K)^+] = D_{\text{USD}}(T)\text{USDJPY}_0 E^{Q_{\text{USD}}}[(\text{EURUSD}_T - K \text{JPYUSD}_T)^+]
$$
where $Q_{\text{JPY}}$ is the JPY forward measure and $Q_{\text{USD}}$ is the USD forward measure.
Next you can use a copula function to mix the marginals of $\text{EURUSD}_T$ and $\text{JPYUSD}_T$ under $Q_{\text{USD}}$:
- You obtain the marginal of $\text{EURUSD}_T$ by differentiating with respect to strike a call option on $\text{EURUSD}$, priced from the $\text{EURUSD}_T$ volatility surface.
- likewise you obtain the marginal of $\text{JPYUSD}_T$ by differentiating with respect to strike a call option on $\text{JPYUSD}_T$ with payoff in USD, which is equivalent to a put option on $\text{USDJPY}_T$ with payoff in JPY and inverse strike, priced from the $\text{USDJPY}$ volatility surface
- you assume a correlation $\rho$ between $\text{EURUSD}_T$ and $\text{JPYUSD}_T$, which you plug in your copula function (for instance a Gaussian copula)
You now have a bivariate model to price any $\text{EURJPY}$ option, from which you can infer (by applying inverse Black-Scholes) the $\text{EURJPY}$ volatility surface.
As you can see the methodology relies strongly on the estimation for the correlation $\rho$.
In fact, it often works the reverse way: from the 3 volatility surfaces for $\text{EURUSD}$, $\text{USDJPY}$ and $\text{EURJPY}$ and the approach above, one can infer an implied correlation $\rho$, and use this correlation for other purposes, such as computing the quanto drift adjustment for a quanto $\text{EURUSD}$ option with payoff in JPY.
There are some papers on this methodology. See for instance http://eprints.lancs.ac.uk/45612/1/10.pdf.