If $x_t$ is a series of logarithmic returns, the standard GARCH(1,1) model for it takes the form
$$
\begin{aligned}
x_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 x_{t-1}^2 + \beta_1 \sigma_{t-1}^2,
\end{aligned}
$$
where $\varepsilon_t$ is $i.i.d.(0,1)$ random variable. (It is straightforward to extend the model order to obtain GARCH($p$,$q$).) If you care about the volatility (measured by standard deviation or variance) of the returns, no absolute values get involved.
A GARCH model that involves absolute values is absolute value GARCH (AVGARCH or TS-GARCH) due to Taylor and Schwert (Bollerslev, 2009, p. 30) and is formulated as follows:
$$
\begin{aligned}
x_t &= \sigma_t \varepsilon_t, \\
\sigma_t &= \omega + \alpha_1 |x_{t-1}| + \beta_1 \sigma_{t-1}.
\end{aligned}
$$
The model is less sensitive to large errors as compared to the regular GARCH model.
References
- Bollerslev, Tim. "Glossary to ARCH (GARCH)." Volatility and Time Series Econometrics: Essays in Honour of Robert F. Engle. 2009.