There is limited data from Arabic-speaking countries on risk factors for depression and anxiety during the COVID-19 pandemic. Country-specific data is necessary given differences in culture, demographics, and COVID-19 infection and mortality rates. The main purpose of the study is to identify the factors associated with symptoms of depression-anxiety in the adult population of Qatar during the first year of the COVID-19 pandemic. We conducted a cross-sectional online survey in Qatar between July and December 2020 after Qatar’s first COVID-19 wave and before the beginning of the second wave. Depression-anxiety was defined as a cut-off of 20 or higher on the Patient Health Questionnaire-Anxiety Depression Scale (PHQ-ADS). Of 1138 participants, 71.0% were female, 69.0% Arabs, 70.0% Non-Qataris, and 77.0% were < 40 years (the median age in Qatar is 32 years). In a fully-adjusted model, six variables were significantly associated with moderate-to-severe levels of depression or anxiety on the PHQ-ADS; Arab ethnicity (OR = 1.67, p = 0.026), never married versus married (OR = 1.69, p = 0.015), prior history of psychiatric disorder versus no history (OR = 1.80, p = 0.009), increased worries due to social media use for COVID-related news/updates (OR = 1.72, p = 0.003), a history of COVID-19 (OR = 1.76, p = 0.039), loneliness (OR = 1.91, p < 0.001), and lower levels of religiosity (OR = 0.96, p = 0.039). These associations also pertained in the reduced model, with the exception of religiosity which was only marginally statistically significant (OR = 0.97, p = 0.055). The potential risk factors identified may assist with anxiety and depression prevention in future COVID-19 waves, and similar national events, and assist with early intervention to treat sufferers.