The improved in vitro regulation of human embryonic stem cell (hESC) pluripotency and differentiation trajectories is required for their promising clinical applications. The temporal and spatial quantification of the molecular interactions controlling pluripotency is also necessary for the development of successful mathematical and computational models. Here we use time-lapse experimental data of OCT4-mCherry fluorescence intensity to quantify the temporal and spatial dynamics of the pluripotency transcription factor OCT4 in a growing hESC colony in the presence and absence of BMP4. We characterise the internal self-regulation of OCT4 using the Hurst exponent and autocorrelation analysis, quantify the intra-cellular fluctuations and consider the diffusive nature of OCT4 evolution for individual cells and pairs of their descendants. We find that OCT4 abundance in the daughter cells fluctuates sub-diffusively, showing anti-persistent self-regulation. We obtain the stationary probability distributions governing hESC transitions amongst the different cell states and establish the times at which pro-fate cells (which later give rise to pluripotent or differentiated cells) cluster in the colony. By quantifying the similarities between the OCT4 expression amongst neighbouring cells, we show that hESCs express similar OCT4 to cells within their local neighbourhood within the first two days of the experiment and before BMP4 treatment. Our framework allows us to quantify the relevant properties of proliferating hESC colonies and the procedure is widely applicable to other transcription factors and cell populations.