We present the first satellite system of the Large Binocular Telescope Satellites Of Nearby Galaxies Survey (LBT-SONG), a survey to characterize the close satellite populations of Large Magellanic Cloud to Milky Way-mass, star-forming galaxies in the Local Volume. In this paper, we describe our unresolved diffuse satellite finding and completeness measurement methodology and apply this framework to NGC 628, an isolated galaxy with $sim1/4$ the stellar mass of the Milky Way. We present two new dwarf satellite galaxy candidates: NGC 628 dwA, and dwB with $text{M}_{text{V}}$ = $-12.2$ and $-7.7$, respectively. NGC 628 dwA is a classical dwarf while NGC 628 dwB is a low-luminosity galaxy that appears to have been quenched after reionization. Completeness corrections indicate that the presence of these two satellites is consistent with CDM predictions. The satellite colors indicate that the galaxies are neither actively star-forming nor do they have the purely ancient stellar populations characteristic of ultrafaint dwarfs. Instead, and consistent with our previous work on the NGC 4214 system, they show signs of recent quenching, further indicating that environmental quenching can play a role in modifying satellite populations even for hosts smaller than the Milky Way.
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