In Viticulture, visual inspection of the plant is a necessary task for measuring relevant variables. In many cases, these visual inspections are susceptible to automation through computer vision methods. Bud detection is one such visual task, central for the measurement of important variables such as: measurement of bud sunlight exposure, autonomous pruning, bud counting, type-of-bud classification, bud geometric characterization, internode length, bud area, and bud development stage, among others. This paper presents a computer method for grapevine bud detection based on a Fully Convolutional Networks MobileNet architecture (FCN-MN). To validate its performance, this architecture was compared in the detection task with a strong method for bud detection, the scanning windows with patch classifier method, showing improvements over three aspects of detection: segmentation, correspondence identification and localization. In its best version of configuration parameters, the present approach showed a detection precision of $95.6\%$, a detection recall of $93.6\%$, a mean Dice measure of $89.1\%$ for correct detection (i.e., detections whose mask overlaps the true bud), with small and nearby false alarms (i.e., detections not overlapping the true bud) as shown by a mean pixel area of only $8\%$ the area of a true bud, and a distance (between mass centers) of $1.1$ true bud diameters. We conclude by discussing how these results for FCN-MN would produce sufficiently accurate measurements of variables bud number, bud area, and internode length, suggesting a good performance in a practical setup.