We develop a novel source coding strategy for sampling and monitoring of a Wiener process. For the encoding process, we employ a four level ``quantization'' scheme, which employs monotone function thresholds as opposed to fixed constant thresholds. Leveraging the hitting times of the Wiener process with these thresholds, we devise a sampling and encoding strategy which does not incur any quantization errors. We give analytical expressions for the mean squared error (MSE) and find the optimal source code lengths to minimize the MSE under this monotone function threshold scheme, subject to a sampling rate constraint.