Heartbeats and a blink are two major physiological signals that provide crucial information on our stress and drowsiness. Hence, it is highly demanded to detect simultaneously heartbeats and blinks in many applications. A Doppler sensor could be a key device to facilitate the non-contact heartbeat and blink detection in daily life. Although many Doppler sensor-based heartbeat and blink detection methods have been independently proposed, when heartbeats and blinks are detected simultaneously with one Doppler sensor, the detection accuracies of such heartbeat and blink detection methods get degraded because of at least two issues: (i) the low SNR (Signal-to-Noise Ratio) of each signal reflected from a subject's chest and face, and (ii) the similarity of the spectrum distribution of heartbeats and a blink. In this paper, we propose a spectrogram-based simultaneous heartbeat and blink detection using one Doppler sensor. In the proposed method, to extract the spectra that might be due to heartbeats and a blink, the spectra on a spectrogram are integrated. Blink detection is then performed by classifying the peaks of the integrated spectrum into a peak due to a blink or a non-blink based on a supervised machine learning classifier trained with a set of the time domain and the time-frequency domain features. Based on the non-blink peaks, heartbeats are detected considering the RRI (R-R Interval) estimated before the investigated peak to prevent the incorrect heartbeat detection. Through the experiments in the case where microwaves are transmitted from one Doppler sensor toward a body including a chest and a face, our proposal has been shown to be able to simultaneously detect heartbeats and blinks with high accuracy.