Besides the up and down quark, what other quarks are present in daily matter around us?

Every nucleon has what are called sea quarks in it, in addition to the valence quarks that define the nucleon as a proton or neutron. Some of those sea quarks, especially the strange quarks, have some secondary relevance in practical terms regarding how the residual strong nuclear force between protons and neutrons in an atomic nucleus is calculated from first principles and how stable a free neutron is if you calculate that from first principles. Strange quarks are also found in the $\Lambda^0$ baryon (which has quark structure $uds$), which is present at a low frequency in cosmic rays, but has a mean lifetime of only about two tenths of a nanosecond and is only indirectly detected in the form of its decay products.

Strange quarks are also relevant at a philosophical level that could impact your daily life, because mesons including strange quarks called kaons, are the lightest and most long lived particles in which CP violation is observed; thus, strange quarks are what made it possible for us to learn that the laws of physics at a quantum level are not independent of an arrow of time.

You could do a lot of sophisticated engineering for a lifetime without ever knowing that second or third generation quarks existed, even nuclear engineering. Indeed, the basic designs of most nuclear power plants and nuclear weapons in use in the United States today were designed before scientists knew that they existed. The fact that protons and neutrons are made out of quarks was a conclusion reached in the late 1960s and not widely accepted until the early 1970s, although strange quark phenomena were observed in high energy physics experiments as early as the 1950s. Third generation fermions were discovered even later. The tau lepton was discovered in 1974, the tau neutrino in 1975, the b quark in 1977, and the top quark in 1995 (although its existence was predicted and almost certain in the 1970s).

Otherwise, these quarks are so ephemeral and require such concentrated energy to produce, that they have no real impact on daily life and are basically never encountered outside of high energy physics experiments, although some of them may be present in and influence to properties of distant neutron stars. Second and third generation quarks also definitely played an important part in the process of the formation of our universe shortly after the Big Bang.

The only second or third generation fermion in the Standard Model with significant practical engineering applications and an impact on daily life and on technologies that are used in the real world are muons (the second generation electron). Muons are observed in nature in cosmic rays (a somewhat misleading term since it doesn't include only photons) and in imaging technologies similar to X-rays but with muons instead of high energy photons. Muons are also used in devices designed to detect concealed nuclear isotypes. Muons were discovered in 1937, although muon neutrinos were first distinguished from electron neutrinos only in 1962, and the fact that neutrinos have mass and that different kinds of neutrinos have different masses was only established experimentally in 1998.

There are all sorts of quarks around us all the time, but they only exist for short periods of time. The up and down quarks are what are known as first generation quarks. They have a relatively small rest mass of $1.7\,\mathrm{MeV}$ and $4.1\,\mathrm{MeV}$ respectively. Comparing this with the 2nd generation of quarks, the strange and charm quarks with masses of $101\,\mathrm{MeV}$ and $1270\,\mathrm{MeV}$, we can see that these are a lot heavier and require a lot more energy to produce. The third generation of quarks, the top and bottom quarks, are over 1000 times heavier than the first generation, with masses of $172\,\mathrm{GeV}$ and $4\,\mathrm{GeV}$ respectively. The heavier a quark is the more energy it takes to produce, and the more quickly particles composed of these quarks will decay into particles with a lower energy.

However that doesn't mean they don't occur naturally. Like the previous answer said, some of them may occur in extreme natural physics like neutron stars. Other than that we only really encounter the heavier 3rd generation quarks in high energy physics experiments. Even then the top quark decays so quickly that it does not exist for long enough for the strong interaction to force it to form a particle.

Particles formed with second generation quarks occur more regularly. For example the $\Lambda^0$ baryon has quark structure $uds$ and was detected in cosmic ray experiments, meaning these particles are created all the time when radiation such as protons from the sun interact with particles in Earth's upper atmosphere. Again it should be noted that these particles then not to exist for very long. I believe the lifetime of the $\Lambda^0$ baryon, for example, is less than a nanosecond.

We mostly observe protons and neutrons in every day life because their quark structure, being made of only the lightest quarks, means they exist for a very long time. Indeed there is still some debate as to how long it would take for protons to decay, as different alternatives to the standard model give different predictions. I believe the proton is thought to have a lifetime of at least $10^{32}$ years.