Patients affected by Attention-Deficit/Hyperactivity Disorder (ADHD) are characterized by impaired executive functioning and/or attention deficits. Our study aim is to determine whether the outcomes measured by the Attention Network Task (ANT), i.e., the reaction times (RTs) to specific target and cue conditions and alerting, orienting, and conflict (or executive control) effects are affected by cognitive training with a Dual n-back task. We considered three groups of young adult participants: ADHD patients without medication (ADHD), ADHD with medication (MADHD), and age/education-matched controls. Working memory training consisted of a daily practice of 20 blocks of Dual n-back task (approximately 30 min per day) for 20 days within one month. Participants of each group were randomly assigned into two subgroups, the first one with an adaptive mode of difficulty (adaptive training), while the second was blocked at the level 1 during the whole training phase (1-back task, baseline training). Alerting and orienting effects were not modified by working memory training. The dimensional analysis showed that after baseline training, the lesser the severity of the hyperactive-impulsive symptoms, the larger the improvement of reaction times on trials with high executive control/conflict demand (i.e., what is called Conflict Effect), irrespective of the participants' group. In the categorical analysis, we observed the improvement in such Conflict Effect after the adaptive training in adult ADHD patients irrespective of their medication, but not in controls. The ex-Gaussian analysis of RT and RT variability showed that the improvement in the Conflict Effect correlated with a decrease in the proportion of extreme slow responses. The Dual n-back task in the adaptive mode offers as a promising candidate for a cognitive remediation of adult ADHD patients without pharmaceutical medication.