In this paper, we propose using a generic protograph to design joint source-channel codes (JSCCs). We present a generalized algorithm, called protograph extrinsic information transfer for JSCC algorithm (PEXIT-JSCC algorithm), for analyzing the channel threshold of the proposed JSCC. We also propose a source generic protograph EXIT (SGP-EXIT) algorithm, which is more appropriate than the generalized source protograph extrinsic information transfer (GSP-EXIT) algorithm, for evaluating the source threshold of a generic protograph. Moreover, a collaborative optimization method based on the SGP-EXIT and PEXIT-JSCC algorithms is proposed to construct generic-protograph JSCCs with good source and channel thresholds. Finally, we construct generic-protograph JSCCs, analyze their decoding thresholds, and compare their theoretical and error performance with JSCC systems based on optimized double-protographs. Results show that our proposed codes can attain channel thresholds within 1 dB from the Shannon limit and outperform double-protograph-based JSCCs.