Yes, you can use 6 ohm and 8 ohm speakers together. However, they may not perform optimally as the amplifier will deliver power unevenly, which could potentially cause damage over time.
Dependence upon speaker impedance ratings and power ratings will ultimately decide if 6 ohm and 8 ohm speakers can coexist together or not. An impedance rating measures the ease of an electric signal passing through your speaker, in this case music. All speakers typically feature either 4, 6, or 8 as their impedance ratings, the lower being easier for amplifiers to drive while higher values present challenges that could damage equipment.
Modern amplifiers are typically capable of handling various impedances and contain protection circuitry to avoid overheating and damage to the amplifier. Unfortunately, mixing a pair of 6-ohm speakers with an amplifier rated for 8-ohms creates too low an impedance level and forces it to struggle to provide sufficient power resulting in distorted audio and weaker overall sound quality.
As such, many individuals opt to use two separate pairs of speakers when using an amplifier that only supports 8-ohm loads; this ensures there is sufficient power available for driving the speakers without overheating or distortion occurring.
Impedance differences between 6 ohm and 8 ohm speakers can also alter their sound quality subtly; music often features multiple frequencies that peak at different times, creating some degree of phase-shift between channels that can sound unnatural and artificial.
Movies tend to present audio in more linear and consistent fashion, creating an authentic sound with less phasing.
Given this information, if you are using all 6 ohm speakers it is generally advised that if using the 8-ohm setting on your receiver as this will likely work without issue. Also bear in mind that most of us tend to crank our amps louder than would be used for professional systems which reduces headroom for speakers and increases chances of them struggling.