This paper focuses on channel estimation for mmWave MIMO systems with 1-bit spatial sigma-delta analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). The channel estimation performance with 1-bit spatial sigma-delta modulators (i.e., ADCs or DACs) depends on the quantization noise modeling. Therefore, we present a new method for modeling the quantization noise by leveraging the deterministic input-output relation of the 1-bit spatial sigma-delta modulator. Using this new noise model, we propose a channel estimation algorithm for a narrowband single-user mmWave line-of-sight MIMO system by determining the unknown angles and path attenuation that characterize the flat-fading channel. Through simulations, we demonstrate that the performance of the developed method is comparable to the traditional unquantized system and significantly better than the conventional 1-bit quantized system.