Probably not, but there are several ways that this could happen.
Severe solar storms can, and do, affect the power grid. They can trigger actual black outs like the one that happened in Quebec in March 1989, and they can cause temporary instabilities in the power levels. These are not the 'spikey' kinds of events that most computer power supplies can protect against, but long-term surges and sags lasting several minutes or hours. If transformers get over heated from the induced DC currents produced by geomagnetic storms, then you could end up with localized problems in the grid that supplies your neighborhood. In Canada they have lots of problems with geomagnetic storm currents that get induced into their power grid and local brownouts are rather common during strong auroral displays. In the United States, though, we rarely have these events in our electrical system.
Another channel is in direct radiation damage to specific memory locations. Solar storms enhance the radiation levels in space near the earth, and these particles when the enter the atmosphere produce 'air showers' of secondary particles...usually neutrons...which can penetrate down to 35,000 feet and even ground level. If you were flying in a commercial jet over, say, the polar route, you would get about one chest X-ray of additional radiation dosage from these neutrons. They also penetrate into avionics and computer circuitry and cane cause temporary 'glitches' as data bits get switched from 'ones' to 'zeros' or the reverse. Although some critical avionics systems have multiple back-ups to protect from these glitches, no one has ever studied how lap-top computers are affected. At ground level, there is substantially less risk from such glitches, but the risk factor isn't zero to 50 decimal places either. I, personally, would not worry about this kind of problem.
Finally, many computers are connected to remote systems via satellite data channels. Satellites do get affected by solar storms, and during a 'big one' you can expect your link to become less reliable and even shut-down completely if the satellite is temporarily disabled. This could cause your computer to 'crash' depending on what it was doing.
I would rate the satellite data link problem as a higher likelihood than the other two possibilities, but for most people not really that big of a risk factor unless you were doing something that demanded 100 percent reliability in a, say, life-critical situation.
All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the
NASA IMAGE/POETRY project.