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, Scanning Windows (SW) based on a patch classifier, showing improvements over three aspects of detection: segmentation, correspondence identification and localization. The best version of FCN-MN showed a detection F1-measure of $88.6\%$ (for true positives defined as detected components whose intersection-over-union with the true bud is above $0.5$), and false positives that are small and near the true bud. Splits -- false positives overlapping the true bud -- showed a mean segmentation precision of $89.3\% (21.7)$, while false alarms -- false positives not overlapping the true bud -- showed 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. The paper concludes by discussing how these results for FCN-MN would produce sufficiently accurate measurements of bud variables such as bud number, bud area, and internode length, suggesting a good performance in a practical setup.