Multivariate motif discovery aims to identify frequently occurring subsequences within multi-dimensional time series, which is a critical machine learning task with wide applications. However, previous motif discovery algorithms often miss complex multivariate motifs and struggle with high computational costs as data scale and dimensionality grow. We propose a novel \underline{L}earnable \underline{M}ultivari\underline{A}te matrix \underline{P}rofile method (L-MAP) that captures inter-dimensional dependencies for comprehensive analysis of multivariate time series. The time series is partitioned into subsequences using the Fourier transform in the frequency domain, with locality-sensitive hashing (LSH) assigning them to buckets based on distinct patterns. Each subsequence is modeled as a graph for multivariate fusion, where triplet learning is used to capture cross-dimensional relationships and form graph embeddings. Unlike prior methods relying on Euclidean distance modeling, our graph-based approach computes all-pairs similarity in a latent space, which constructs the multivariate matrix profile from distributions formed by embedding clusters. Extensive experiments on multivariate datasets from diverse domains demonstrate that L-MAP outperforms state-of-the-art methods in motif discovery, offering superior quality, diversity, and scalability efficiency.