Memristive devices being applied in neuromorphic computing are envisioned to significantly improve the power consumption and speed of future computing platforms. The materials used to fabricate such devices will play a significant role in their viability. Graphene is a promising material, with superb electrical properties and the ability to be produced sustainably. In this paper, we demonstrate that a fabricated graphene-pentacene memristive device can be used as synapses within Spiking Neural Networks (SNNs) to realise Spike Timing Dependent Plasticity (STDP) for unsupervised learning in an efficient manner. Specifically, we verify operation of two SNN architectures tasked for single digit (0-9) classification: (i) a simple single-layer network, where inputs are presented in 5x5 pixel resolution, and (ii) a larger network capable of classifying the Modified National Institute of Standards and Technology (MNIST) dataset, where inputs are presented in 28x28 pixel resolution. Final results demonstrate that for 100 output neurons, after one training epoch, a test set accuracy of up to 86% can be achieved, which is higher than prior art using the same number of output neurons. We attribute this performance improvement to homeostatic plasticity dynamics that we used to alter the threshold of neurons during training. Our work presents the first investigation of the use of green-fabricated graphene memristive devices to perform a complex pattern classification task. This can pave the way for future research in using graphene devices with memristive capabilities in neuromorphic computing architectures. In favour of reproducible research, we make our code and data publicly available https://anonymous.4open.science/r/c69ab2e2-b672-4ebd-b266-987ee1fd65e7.