No, it seems that the 528hz claim is either completely made up, or at best based off of recordings artificially adjusted to be heard by humans. The Sun simply doesn't oscillate that quickly or that consistently for the claim to be believable.
Why 528hz is not realistic: Helioseismology is both an awesome word and the research field concerned with studying the internal structure of the Sun through its oscillations. These oscillations operate as fairly regular waves that travel throughout the sun, and 'fairly regular waves' is essentially what sound is.
However, as explained at this Stanford site(careful, it automatically downloads a sound file of the Sun for some reason), there are many overlapping waves moving through the Sun and they take hours to complete a cycle.
Because of the time it takes the waves to travel through the interior of the sun, the oscillations are measured in the milli- or micro-Hertz, several orders of magnitude smaller that 528hz. This paper describes how the largest interior solar oscillations aren't more than a few millihertz:
The Sun's oscillations are observed in line-of-sight Doppler velocity measurements over the visible solar disk, and also in measurements of variations of its continuum intensity of radiation caused by compression of the radiating gas by the waves... The largest-amplitude modes have frequencies of around 3 mHz.
This paper gives similar values in its abstract:
...from 1,000 to 4,000 µHz, the spectral lines closely match the frequencies of known solar pressure modes.
To reach a 528hz frequency, these waves would have to complete their trip through the Sun 528 times per second, not once every few hours. I haven't done the math, but I assume a million kilometers per second is a bit above the speed of sound within the Sun.
Possible origin of the claim: I couldn't find any NASA papers discussing 528hz and the Sun. The only potential source of the claim that I could find are videos of sped-up recordings of these waves, such as this video. Part of the first sound clip does sound a bit like 528hz(it's basically that classic 'Om' sound you hear from people meditating). However, as explained on the Standford site, these 'sound recordings' are weeks worth of oscillations compressed into a few seconds of video, and it's a small segment of the many, many waves travelling through the Sun at any time.
Considering the other flowery, exaggerated, and unsourced claims on the site, it can be safely assumed that this claim is equally without merit.